provided the tuning parameter \(V\)  in our estimator satisfies \(V \ge V_{\mathrm {mars}}(f^*)\) . The multiplicative constant underlying \(O(\cdot )\)  in
(REF ) depends on
\(d, \sigma \) , and \(V\)  (see Theorem REF  for the explicit
expression). This result is interesting from the following
perspective. Already in the univariate case (\(d = 1\) ), the estimator
\(\hat{f}_{n, V}^1\)  achieves the rate \(n^{-4/5}\)  (see, e.g.,
[1]} or
[2]}). Thus,
(REF ) implies that in going from the univariate to
the multivariate setting, the rate only deteriorates by a logarithmic
multiplicative factor, which suggests that our LASSO method for
MARS fitting can be an effective function estimation technique in
higher dimensions. The assumption that the design points \(x^{(1)},\dots , x^{(n)}\)  form an equally spaced lattice in \([0, 1]^d\)  is
admittedly restrictive, although this assumption is standard in
theoretical analysis in nonparametric function estimation (see,
e.g., [3]}), and it is also directly relevant to
applications involving image denoising. However, we are currently unable to
handle more general design assumptions.
